On sums of Hecke series in short intervals
Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 453-468.

On a ∑ K-G≤k j ≤K+G α j H j 3 (1 2)≪ ϵ GK 1+ϵ pour K ϵ ≤G≤K,ouα j =ρ j (1) 2 (coshπk j ) -1 ,etρ j (1) est le premier coefficient de Fourier de forme de Maass correspondant à la valeur propre λ j =k j 2 +1 4 à laquelle le série de Hecke H j (s) est attachée. Ce résultat fournit l’estimation nouvelle H j ( 1 2 ) ≪ ϵ k j 1 3 + ϵ .

We have ∑ K-G≤k j ≤K+G α j H j 3 (1 2)≪ ϵ GK 1+ϵ for K ϵ ≤G≤K,whereα j =ρ j (1) 2 (coshπk j ) -1 ,andρ j (1) is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue λ j =k j 2 +1 4 to which the Hecke series H j (s) is attached. This result yields the new bound H j ( 1 2 ) ≪ ϵ k j 1 3 + ϵ . 

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     title = {On sums of {Hecke} series in short intervals},
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     publisher = {Universit\'e Bordeaux I},
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     year = {2001},
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Ivić, Aleksandar. On sums of Hecke series in short intervals. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 453-468. http://archive.numdam.org/item/JTNB_2001__13_2_453_0/

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